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Properties

Label 15.5.52657870336...0000.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,2^{15}\cdot 5^{6}\cdot 23^{5}\cdot 59^{2}\cdot 157^{2}\cdot 38707^{4}\cdot 910849^{2}$
Root discriminant $3814.44$
Ramified primes $2, 5, 23, 59, 157, 38707, 910849$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T97

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19817984000, 0, 25020204800, -19260603200, -8249235840, 12454039840, -4433943936, -405543860, 191481624, -23582115, -541864, 237803, -1152, -792, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 792*x^13 - 1152*x^12 + 237803*x^11 - 541864*x^10 - 23582115*x^9 + 191481624*x^8 - 405543860*x^7 - 4433943936*x^6 + 12454039840*x^5 - 8249235840*x^4 - 19260603200*x^3 + 25020204800*x^2 - 19817984000)
 
gp: K = bnfinit(x^15 - 792*x^13 - 1152*x^12 + 237803*x^11 - 541864*x^10 - 23582115*x^9 + 191481624*x^8 - 405543860*x^7 - 4433943936*x^6 + 12454039840*x^5 - 8249235840*x^4 - 19260603200*x^3 + 25020204800*x^2 - 19817984000, 1)
 

Normalized defining polynomial

\( x^{15} - 792 x^{13} - 1152 x^{12} + 237803 x^{11} - 541864 x^{10} - 23582115 x^{9} + 191481624 x^{8} - 405543860 x^{7} - 4433943936 x^{6} + 12454039840 x^{5} - 8249235840 x^{4} - 19260603200 x^{3} + 25020204800 x^{2} - 19817984000 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-526578703368934513642333503864526678250539093504000000=-\,2^{15}\cdot 5^{6}\cdot 23^{5}\cdot 59^{2}\cdot 157^{2}\cdot 38707^{4}\cdot 910849^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3814.44$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 23, 59, 157, 38707, 910849$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} + \frac{3}{8} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{10} - \frac{5}{16} a^{6} - \frac{1}{2} a^{5} + \frac{5}{16} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{160} a^{11} + \frac{1}{20} a^{9} + \frac{1}{20} a^{8} - \frac{37}{160} a^{7} + \frac{7}{20} a^{6} - \frac{7}{32} a^{5} - \frac{1}{10} a^{4} + \frac{3}{8} a^{3} - \frac{7}{20} a^{2} - \frac{1}{2} a$, $\frac{1}{320} a^{12} + \frac{1}{40} a^{10} + \frac{1}{40} a^{9} - \frac{37}{320} a^{8} + \frac{7}{40} a^{7} + \frac{25}{64} a^{6} - \frac{1}{20} a^{5} + \frac{3}{16} a^{4} - \frac{7}{40} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{6400} a^{13} + \frac{1}{800} a^{11} + \frac{3}{400} a^{10} + \frac{203}{6400} a^{9} + \frac{67}{800} a^{8} + \frac{217}{1280} a^{7} + \frac{53}{800} a^{6} - \frac{153}{320} a^{5} + \frac{179}{400} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{4} a$, $\frac{1}{14649336224913012978759200960478089343171460609389444565337676800} a^{14} - \frac{154012777219111395074455224172653912947968400079618370877393}{3662334056228253244689800240119522335792865152347361141334419200} a^{13} - \frac{2768999303376315933427142259799395728652326142954376421337489}{1831167028114126622344900120059761167896432576173680570667209600} a^{12} + \frac{1501348339616729462912533680424498133081134162438660681027}{57223969628566456948278128751867536496763518005427517833350300} a^{11} + \frac{243003446571894493879221296249089853937265990861063181238078187}{14649336224913012978759200960478089343171460609389444565337676800} a^{10} + \frac{20641601274314422679421808680601891967455729100809334982423839}{732466811245650648937960048023904467158573030469472228266883840} a^{9} - \frac{883830442356592760094910063460018983454826901577179627272027347}{14649336224913012978759200960478089343171460609389444565337676800} a^{8} - \frac{33551089998207848500794418556425572766745111984975039093208821}{192754424012013328667884223164185386094361323807755849543916800} a^{7} - \frac{1115919807733971571555591504669944964721004529783064659526001937}{3662334056228253244689800240119522335792865152347361141334419200} a^{6} + \frac{13534176412482629044841113171573836137281379556310978238301519}{915583514057063311172450060029880583948216288086840285333604800} a^{5} + \frac{58486418574077651273907967133811396949458984661524088065310131}{457791757028531655586225030014940291974108144043420142666802400} a^{4} + \frac{88650427328401985001801657804792510610403364641245823608389}{2288958785142658277931125150074701459870540720217100713334012} a^{3} - \frac{11673250051692688737371970197053712748637352377120844618074953}{45779175702853165558622503001494029197410814404342014266680240} a^{2} + \frac{12368557120084992379799369485263548051541801790060250939756}{572239696285664569482781287518675364967635180054275178333503} a + \frac{133586024884113213429156738951023720180128456443914301928448}{572239696285664569482781287518675364967635180054275178333503}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4391960199450000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T97:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2592000
The 70 conjugacy class representatives for [A(5)^3:2]S(3) are not computed
Character table for [A(5)^3:2]S(3) is not computed

Intermediate fields

3.1.23.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 siblings: data not computed
Degree 45 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ R

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.4$x^{6} + x^{2} + 1$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.6.9.6$x^{6} + 4 x^{2} + 8$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59Data not computed
$157$$\Q_{157}$$x + 5$$1$$1$$0$Trivial$[\ ]$
157.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
157.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
157.4.2.2$x^{4} - 157 x^{2} + 147894$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
157.6.0.1$x^{6} - x + 61$$1$$6$$0$$C_6$$[\ ]^{6}$
38707Data not computed
910849Data not computed